# Derivation of Young modulus in highly porous foam

Here we estimate the Young modulus of an open-cell foam, represented schematically as shown below, where each cubic cell has a side length *l* and is made of struts with a square cross-section of thickness *t*, whose Young modulus is *E*_{S}. The derivation is similar to that used for the honeycomb. However because the cell is cubic rather than hexagonal, θ = 0, giving the deflection of a single half-beam, δ, as

\[\delta = \frac{1}{{24}}\frac{{F{l^3}}}{{{E_{\rm{S}}}I}}\]

where *I* is the second moment of area, where *I* = *t*^{4}/12. This gives the deflection in the loading direction, Δ*x*, as

\[\Delta x = 2\delta = \frac{F}{{{E_{\rm{S}}}t}}{\rm{ }}{\left( {\frac{l}{t}} \right)^3}\]

This gives the strain, ε, as

\[\varepsilon = \frac{F}{{{E_{\rm{S}}}}}{\rm{ }} \cdot \frac{{{l^2}}}{{{t^4}}}\]

The stress, σ, arising from the applied force *F* is

\[\sigma = \frac{F}{{{l^2}}}\]

This gives an expression for the Young modulus of the open-cell foam, *E*

\[E = {E_{\rm{S}}}{\rm{ }}{\left( {\frac{t}{l}} \right)^4}\]

Now

\[\frac{\rho }{{{\rho _{\rm{S}}}}} \propto {\left( {\frac{t}{l}} \right)^2}\]

The relative elastic modulus of the open-cell foam, *E*/*E*_{S}, is therefore related to the relative density, ρ/ρ_{S}, according to

\[\frac{E}{{{E_{\rm{S}}}}} = k{\rm{ }}{\left( {\frac{\rho }{{{\rho _{\rm{S}}}}}} \right)^2}\]

where *k* is a constant approximately equal to 1.